{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Quantization of Signals\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Non-Linear Requantization of a Speech Signal\n",
    "\n",
    "Speech signals have a [non-uniform amplitude distribution](../random_signals/important_distributions.ipynb#Example) which is often modeled by the Laplace distribution. Linear uniform quantization is not optimal for speech signals, since small signal amplitudes are more likely than higher ones. This motivates a non-linear quantization scheme, where the signal is companded before linear quantization and expanded afterwards. \n",
    "\n",
    "The following example illustrates the [A-law quantization scheme](https://en.wikipedia.org/wiki/A-law_algorithm) used in European telephone networks. In this scheme the signal is first companded by the non-linear A-law characteristic before a linear uniform quantizer is used. This results overall in a non-linear quantization characteristic. First some functions for A-law companding/expanding, quantization and evaluation are defined."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import soundfile as sf\n",
    "%matplotlib inline\n",
    "\n",
    "\n",
    "def A_law_compander(x):\n",
    "    A = 87.6\n",
    "    y = np.zeros_like(x)\n",
    "    idx = np.where(np.abs(x) < 1/A)\n",
    "    y[idx] = A*np.abs(x[idx]) / (1 + np.log(A))\n",
    "    idx = np.where(np.abs(x) >= 1/A)\n",
    "    y[idx] = (1 + np.log(A*np.abs(x[idx]))) / (1 + np.log(A))\n",
    "\n",
    "    return np.sign(x)*y\n",
    "\n",
    "\n",
    "def A_law_expander(y):\n",
    "    A = 87.6\n",
    "    x = np.zeros_like(y)\n",
    "    idx = np.where(np.abs(y) < 1/(1+np.log(A)))\n",
    "    x[idx] = np.abs(y[idx])*(1+np.log(A)) / A\n",
    "    idx = np.where(np.abs(y) >= 1/(1+np.log(A)))\n",
    "    x[idx] = np.exp(np.abs(y[idx])*(1+np.log(A))-1)/A\n",
    "\n",
    "    return np.sign(y)*x\n",
    "\n",
    "\n",
    "def uniform_midtread_quantizer(x, w):\n",
    "    # quantization step\n",
    "    Q = 1/(2**(w-1))\n",
    "    # limiter\n",
    "    x = np.copy(x)\n",
    "    idx = np.where(x <= -1)\n",
    "    x[idx] = -1\n",
    "    idx = np.where(x > 1 - Q)\n",
    "    x[idx] = 1 - Q\n",
    "    # linear uniform quantization\n",
    "    xQ = Q * np.floor(x/Q + 1/2)\n",
    "\n",
    "    return xQ\n",
    "\n",
    "\n",
    "def evaluate_requantization(x, xQ):\n",
    "    e = xQ - x\n",
    "    # SNR\n",
    "    SNR = 10*np.log10(np.var(x)/np.var(e))\n",
    "    print('SNR: {:2.1f} dB'.format(SNR))\n",
    "    # normalize error\n",
    "    e = .2 * e / np.max(np.abs(e))\n",
    "    return e"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quantization Characteristic\n",
    "\n",
    "Lets first take a look at the non-linear characteristic of the A-law requantizer. The left plot shows the characteristic of the A-law companding and linear-quantization. The right plot shows the overall characteristic for companding, linear quantization and expansion. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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   ],
   "source": [
    "x = np.linspace(-1, 1, 2**16)\n",
    "y = A_law_compander(x)\n",
    "yQ4 = uniform_midtread_quantizer(y, 4)\n",
    "yQ8 = uniform_midtread_quantizer(y, 8)\n",
    "xQ4 = A_law_expander(yQ4)\n",
    "xQ8 = A_law_expander(yQ8)\n",
    "\n",
    "plt.figure(figsize=(10, 4))\n",
    "\n",
    "plt.subplot(121)\n",
    "plt.plot(x, yQ4, label=r'$w=4$ bit')\n",
    "plt.plot(x, yQ8, label=r'$w=8$ bit')\n",
    "plt.title('Compansion and linear quantization')\n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel(r'$x_Q$')\n",
    "plt.legend(loc=2)\n",
    "plt.axis([-1.1, 1.1, -1.1, 1.1])\n",
    "plt.grid()\n",
    "\n",
    "plt.subplot(122)\n",
    "plt.plot(x, xQ4, label=r'$w=4$ bit')\n",
    "plt.plot(x, xQ8, label=r'$w=8$ bit')\n",
    "plt.title('Overall')\n",
    "plt.xlabel(r'$x$')\n",
    "plt.ylabel(r'$x_Q$')\n",
    "plt.legend(loc=2)\n",
    "plt.axis([-1.1, 1.1, -1.1, 1.1])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Signal-to-Noise Ratio\n",
    "\n",
    "Now the signal-to-noise ratio (SNR) is computed for a Laplace distributed signal for various RMS levels $\\sigma_x / x_\\mathrm{min}$. The results show that the non-linear quantization scheme provides a constant SNR over a wide range of levels. The SNR is additional higher as for [linear quantization](../quantization/linear_uniform_quantization_error.ipynb#Example) of a Laplace distributed signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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       "<Figure size 768x384 with 1 Axes>"
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   ],
   "source": [
    "w = 8  # wordlength of the quantizer\n",
    "A = np.logspace(-50/20, -10/20, num=500)  # relative RMS levels\n",
    "N = int(1e6)  # number of samples\n",
    "np.random.seed(1)\n",
    "\n",
    "\n",
    "def compute_SNR(a):\n",
    "    # compute input signal\n",
    "    x = np.random.laplace(size=N, scale=a/np.sqrt(2))\n",
    "    # quantize signal\n",
    "    y = A_law_compander(x)\n",
    "    yQ = uniform_midtread_quantizer(y, 8)\n",
    "    xQ = A_law_expander(yQ)\n",
    "    e = xQ - x\n",
    "    # compute SNR\n",
    "    SNR = 10*np.log10((np.var(x)/np.var(e)))\n",
    "\n",
    "    return SNR\n",
    "\n",
    "\n",
    "# quantization step\n",
    "Q = 1/(2**(w-1))\n",
    "# compute SNR for given RMS levels\n",
    "SNR = [compute_SNR(a) for a in A]\n",
    "\n",
    "# plot results\n",
    "plt.figure(figsize=(8, 4))\n",
    "plt.plot(20*np.log10(A), SNR)\n",
    "plt.xlabel(r'RMS level $\\sigma_x / x_\\mathrm{min}$ in dB')\n",
    "plt.ylabel('SNR in dB')\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Requantization of a Speech Sample\n",
    "\n",
    "Finally we requantize a speech sample with a linear and the A-law quantization scheme. The speech signal was originally recorded with a wordlength of $w=16$ bits using linear uniform quantization. First the A-law compansion is applied, then quantization by a linear uniform quantizer with a wordlength of $w=8$ bits. This scheme is used in the backbone of many telephone networks resulting in a total bit-rate of 64 kbits/s. Listen to the samples! Note, the quantization error has been normalized."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SNR: 35.7 dB\n",
      "SNR: 38.2 dB\n"
     ]
    }
   ],
   "source": [
    "# load speech sample\n",
    "x, fs = sf.read('../data/speech_8k.wav')\n",
    "x = x/np.max(np.abs(x))\n",
    "\n",
    "# linear quantization\n",
    "xQ = uniform_midtread_quantizer(x, 8)\n",
    "e = evaluate_requantization(x, xQ)\n",
    "sf.write('speech_8k_8bit.wav', xQ, fs)\n",
    "sf.write('speech_8k_8bit_error.wav', e, fs)\n",
    "\n",
    "# A-law quantization\n",
    "y = A_law_compander(x)\n",
    "yQ = uniform_midtread_quantizer(y, 8)\n",
    "xQ = A_law_expander(yQ)\n",
    "e = evaluate_requantization(x, xQ)\n",
    "sf.write('speech_Alaw_8k_8bit.wav', xQ, fs)\n",
    "sf.write('speech_Alaw_8k_8bit_error.wav', e, fs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Original Signal**\n",
    "\n",
    "<audio src=\"../data/speech_8k.wav\" controls>Your browser does not support the audio element.</audio>\n",
    "[../data/speech_8k.wav](../data/speech.wav)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Linear Requantization to $w=8$ bit** \n",
    "\n",
    "Signal\n",
    "\n",
    "<audio src=\"speech_8k_8bit.wav\" controls>Your browser does not support the audio element.</audio>\n",
    "[speech_8k_8bit.wav](speech_8k_8bit.wav)\n",
    "\n",
    "Error\n",
    "\n",
    "<audio src=\"speech_8k_8bit_error.wav\" controls>Your browser does not support the audio element.</audio>\n",
    "[speech_8k_8bit_error.wav](speech_8k_8bit_error.wav)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**A-law Requantization to $w=8$ bit**\n",
    "\n",
    "Signal\n",
    "\n",
    "<audio src=\"speech_Alaw_8k_8bit.wav\" controls>Your browser does not support the audio element.</audio>\n",
    "[speech_Alaw_8k_8bit.wav](speech_Alaw_8k_8bit.wav)\n",
    "\n",
    "Error\n",
    "\n",
    "<audio src=\"speech_Alaw_8k_8bit_error.wav\" controls>Your browser does not support the audio element.</audio>\n",
    "[speech_Alaw_8k_8bit_error.wav](speech_Alaw_8k_8bit_error.wav)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
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